Unformatted text preview: spectrum Of a field K. Thanks to (22.214.171.124.11), the homomorphism (126.96.36.199.1) has source and target of the same dimension, as does (188.8.131.52.2), and both homomorphisms have the same kernel, namely F i+1(-3 fcnon i. Q.E.D. (184.108.40.206) Corollary. Assumptions as in (2.3.2), and n>O fixed, suppose that the map h(i) figuring in (220.127.116.11.4) is an isomorphism. Then: (18.104.22.168.1) The Cs-module u+a~F~ox~-t(R"f,(f2}/s(logD))) is locally free, and its formation commutes with arbitrary change of base S'--~ S. (22.214.171.124.2) The canonical mapping F~on i- 1 RnJ, (f2~c/s (log D)) ,- FZo ~ ' @ F i + 1 ~ FZo- ~ i- 1 is an isomorphism. Proof. It suffices to prove (126.96.36.199.2), since by hypothesis F~o ~ ~-~/F~o: i is locally free, and its formation commutes with arbitrary change of base S'-~ S. The composite mapping, formed from the bottom half of the diagram (188.8.131.52.4), F:o~ '-1 (R" f, (t2~:/s (log D))) hi, +x), R"f, (~]:s (log D))/F '+ 2 F~o-~ '(R"J, (Q'x/s (log D))) ~hc0 -~ R.f, (f2~,/s (log D))/F '+1...
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- Fall '11
- Equivalence relation, Isomorphism, Homomorphism, Congruence relation, Algebraic Solutions of Differential Equations